Introduction to arithmetic sequences | Sequences, series and induction | Precalculus

Understanding Arithmetic Sequences

Arithmetic sequences are a key idea in math where each number in the sequence is a certain amount larger than the one before it. This article will help you understand what makes a sequence arithmetic and how to describe them using two different methods: explicit and recursive.

What is an Arithmetic Sequence?

An arithmetic sequence is a list of numbers where each number is obtained by adding a fixed number, called the common difference, to the previous one. This pattern makes it easy to spot arithmetic sequences.

Identifying Arithmetic Sequences

To see if a sequence is arithmetic, look at the differences between each pair of numbers. For example, consider this sequence:

• Sequence: -5, -3, -1, 1

In this sequence:

• From -5 to -3, we add 2.
• From -3 to -1, we add 2.
• From -1 to 1, we add 2.

Since we add the same number (2) each time, this is an arithmetic sequence.

Explicit and Recursive Definitions

We can describe this arithmetic sequence in two ways:

1. Explicit Definition:
   • Let a_n be the nth term of the sequence. The formula is:
   • a_n = -5 + 2(n – 1)
   • Here, -5 is the first term, and we add 2 for each next term.
2. Recursive Definition:
   • Start with the first term:
   • a₁ = -5
   • For n ≥ 2:
   • a_n = a_n-1 + 2
   • Each term is the previous term plus 2.

Another Example of an Arithmetic Sequence

Let’s look at another sequence:

• Sequence: 100, 107, 114, 121

In this case:

• From 100 to 107, we add 7.
• From 107 to 114, we add 7.
• From 114 to 121, we add 7.

This is also an arithmetic sequence. We can define it like this:

1. Explicit Definition:
   • a_n = 100 + 7(n – 1)
2. Recursive Definition:
   • a₁ = 100
   • a_n = a_n-1 + 7 (n ≥ 2)

General Form of Arithmetic Sequences

In general, an arithmetic sequence can be written as:

• Explicit Form:
  • a_n = k + d(n – 1)
  • where k is the first term and d is the common difference.
• Recursive Form:
  • a₁ = k
  • a_n = a_n-1 + d (n ≥ 2)

Non-Arithmetic Sequences

Not all sequences are arithmetic. For example, consider this sequence:

• Sequence: 1, 3, 6, 10

In this sequence:

• From 1 to 3, we add 2.
• From 3 to 6, we add 3.
• From 6 to 10, we add 4.

Since the amount added changes each time, this is not an arithmetic sequence.

Defining a Non-Arithmetic Sequence

We can still describe this sequence using a recursive method:

1. Base Case:
   • a₁ = 1
2. Recursive Definition:
   • a_n = a_n-1 + (n – 1) (n ≥ 2)

Here, the amount added depends on the position in the sequence, which makes it different from an arithmetic sequence.

Conclusion

Understanding arithmetic sequences is important for learning more advanced math topics. By recognizing their patterns and knowing how to describe them explicitly or recursively, you can easily work with these sequences in different math problems.